[Paper Review] Convergence of the Formal Expansion for lambda_d(p) of the Monomer-Dimer Problem for Small p

Paul Federbush|arXiv (Cornell University)|Jan 24, 2011

Mathematical functions and polynomials4 references4 citations

TL;DR

This paper proves that the formal power series expansion of the monomer-dimer free energy $λ_d(p)$ in powers of $p$, derived from a cluster expansion, converges absolutely for sufficiently small $p$ independent of dimension $d$. The convergence is established via recursive bounds on cluster expansion kernels $\bar{J}_k$ and contraction mapping arguments in a Banach space of sequences, ensuring analyticity of $\lambda_d(p)$ in $p$ near zero for all $d$. The result validates the asymptotic expansion for small monomer densities.

ABSTRACT

Shmuel Friedland and the author recently presented a formal expansion for lambda_d(p) of the monomer-dimer problem. Herein we prove that if the terms in the expansion are rearranged as a power series in p, then for sufficiently small p this series converges.

Motivation & Objective

• To establish the convergence of the formal power series expansion of $\lambda_d(p)$ in powers of $p$ for small $p$, independent of dimension $d$.
• To validate the asymptotic expansion of $\lambda_d(p)$ derived from cluster expansion kernels $\bar{J}_k$ as a convergent series for small $p$.
• To provide a rigorous foundation for the formal asymptotic expansion of the monomer-dimer free energy in the low monomer density regime.

Proposed method

• Derive the formal expansion of $\lambda_d(p)$ as a power series in $p$ by reorganizing the cluster expansion in inverse powers of $d$, yielding $\lambda_d(p) = S + \sum_{s=2}^\infty p^s g_s$.
• Define a Banach space $\mathcal{S}$ of sequences $\alpha_k$ with norms $\|\alpha\| = \sum_k 2^k |\alpha_k|$, and use it to analyze convergence of iterative solutions to the master equation for $\alpha_k$.
• Establish recursive bounds on the cluster expansion kernels $\bar{J}_k$ via a generalized cluster expansion formalism, proving $|\bar{J}_k| \leq (4e)^k$ uniformly in $d$, ensuring uniform decay.
• Use a contraction mapping argument on the iteration $\alpha_k' = \bar{J}_k p^k \cdot \left(1 - 2\sum i\alpha_i\right)^{-2k} \cdot \left(1 - 2\sum i\alpha_i/p\right)^k$ to prove convergence of the series for $\lambda_d(p)$.
• Prove that the mapping defined by the iteration preserves the norm bound $\|\alpha\| \leq \varepsilon p$, ensuring convergence for small enough $p$.
• Leverage the uniform bound on $|\bar{J}_k|$ and the contraction property to show that the resulting series for $\lambda_d(p)$ converges absolutely for $0 \leq p < p_0$, with $p_0$ independent of $d$.

Experimental results

Research questions

• RQ1 Does the formal power series expansion of $\lambda_d(p)$ in $p$ converge for small $p$, uniformly in $d$?
• RQ2 Can the cluster expansion formalism for the monomer-dimer problem be rigorously shown to yield a convergent series in $p$ near zero?
• RQ3 Is the asymptotic expansion of $\lambda_d(p)$ analytic in $p$ for small $p$, regardless of dimension $d$?
• RQ4 What uniform bounds on the cluster expansion kernels $\bar{J}_k$ ensure convergence of the series for $\lambda_d(p)$ across all $d$?
• RQ5 Can the convergence be established via a contraction mapping argument in a suitable Banach space of sequences?

Key findings

• The formal power series expansion of $\lambda_d(p)$ in $p$ converges absolutely for all $d$ when $p < p_0$, where $p_0$ is a positive constant independent of $d$.
• The bound $|\bar{J}_k| \leq (4e)^k$ holds uniformly for all $d$, ensuring uniform decay in the cluster expansion kernels.
• The convergence of the series is proven via a contraction mapping argument in the Banach space $\mathcal{S}$, with $\|\alpha\| \leq \varepsilon p$ ensuring convergence for small $p$.
• The coefficients $a_k(d)$ in the $p$-series expansion of $\lambda_d(p)$ are explicitly computed up to $k=6$, with $a_5(d) = \frac{1}{16d^3} - \frac{39}{640d^4}$ and $a_6(d) = \frac{1}{24d^3} - \frac{1}{32d^4} - \frac{19}{1920d^5}$.
• The result confirms the analyticity of $\lambda_d(p)$ in $p$ near zero for all $d$, validating the use of the formal expansion in the low monomer density regime.
• The methods do not reach the full range $p \in [0,1]$, but the convergence result is robust and suggests deeper analytic structure in $\lambda_d(p)$.